Your compost will freeze and that is okay!
If you're limited to only one bin you need to turn it more often. Not sure why you lined it with fabric but if it's for containment purpose you might need to add a bit more air into the center of the pile.
I have a good sized yard and run three hog wire compost bins that are awesome now that I'm an older slower gardener!
I've been limited to one... so if your yard is small...
As soon as your pile has thawed enough to tear it down: Get yourself a 6" diam PVC pipe that is 6" taller in height than your compost pile and drill a whole bunch of holes into it, pound 2-3, 24" tall survey spikes into the center just to steady that pipe and set that oxygen life giver in the center!
You need to get more air into your pile.
Look at it like a lone tree that simply sheds leaves and collect at the base, it takes longer for the leaves to break down, holds in the water, keeps the worms/bugs/bio moving slower, allows the tree to go slow and steady.
Your compost pile has no upper living branches, not enough light, no air, not enough water, too much water...
Compost, in nature, is a healthy living breathing life all its own.
You want activity and a bit of heat in garden compost. That happens with the seasons but also what you put into it! Do you want a slow compost or an active compost?
Organic chicken manure smells like the worst yack ever but it heats up and you want this at the base and in the outer rim of your layers.
Do not use white office paper or glossy inserts (durr...) smash your egg shells, thin slice your scrap veggies, melon rinds, hook up with your local coffee shack (if you have more grounds than you think your compost can take, throw it on your grass, your shrubs!) No pork or beef bones, no fatty scraps! A dab of butter or olive oil on those table scrap veggies are no big deal! If you fish, (stream or lake...) put those bones in the ground with some fire ash and plant a fruit tree.
You already know to mulch your leaf fall!
Add compost worms when you add a big batch of kitchen cuttings, these are not fishing worms! Red Wigglers are hard working, voracious pooping machines! Keep them to the middle or cooler areas of your pile and away from the glorious chicken manure!
Make sure your grass clippings do not have any commercial sprays, topical fertilizers, dog or cat feces... If you use chemicals or fertilizers on your grass, set up a separate compost and only use this compost on your non-fruit/food plantings. This might gross you out but you, your buddies can pee all over the grass clipping compost & it works incredibly well but not on your food compost pile! yechhh Human pee is huge in some gardening cultures.
Now, you will have had a great work out and depending on the progress of your pile, your other garden tasks, you can plan on turning that pile again at 4-6 weeks or let it slow cook.
NOTE: Only cover the top of your pile if you will have freezing temps, huge snow or excessive rainfall. Your compost requires air, water, light and added nutrition in rotation and is cheaper than gym fees!
Happy Gardening!